IMC 2012 Q16
16. The diagram shows a large equilateral triangle divided by three straight lines into seven regions. The three grey regions are equilateral triangles with sides of length 5 cm and the central black region is an equilateral triangle with sides of length 2 cm.
What is the side length of the original large triangle?
- A 18 cm
- B 19 cm
- C 20 cm
- D 21 cm
- E 22 cm
Intermediate Mathematical Challenge 2012, Question 16. © UK Mathematics Trust. Original paper on ukmt.org.uk
Answer: B 19 cm
As triangle \(ABC\) is equilateral, \(\angle BAC = 60^\circ\). Since the grey triangles are equilateral, \(\angle ADE = 60^\circ\), so the triangle \(ADE\) is equilateral. The length of the side of this triangle is equal to the length of \(DE\) = (5 + 2 + 5) cm = 12 cm. So \(AF = AD - FD\) = (12 − 5) cm = 7 cm. By a similar argument, we deduce that \(BD\) = 7 cm, so the length of the side of triangle \(ABC\) = (7 + 5 + 7) cm = 19 cm.
Solution © UK Mathematics Trust, from the IMC 2012 solutions. Longer versions with investigations: Solutions and investigations (PDF).